Fundamental solutions to Hertzian contact problems at nanoscale

Based on surface elasticity theory, the effects of surface energy on Hertzian contact problems at nanoscale are considered. The complex variable function method is adopted to derive the fundamental solutions to the Hertzian contact problem. As examples, the deformations induced by uniformly distributed traction and concentrated force are analyzed in detail. The results reveal some interesting characteristics in contact mechanics, which are distinctly different from those in classical elasticity theory. At nanoscale, the shear displacement gradient on the deformed surface transits continuously across the uniformly distributed loading boundary as a result of surface effects. In addition, for nano-indentation, the indent depth depends strongly on the surface stress.